You can probably try to show that the set is linearly independent over the field . The result will follow immediately.

It seems to be fairly straightforward if you know some field theory. (But I haven't actually tried it, so I could be wrong.)

I am sorry, it was my bad for not saying. I was aware there was an algebraic proof, but you see my cousin asked me and he's in calc III, I was hoping there may be an "elementary" number theoretic proof. Thanks anyways, I appreciate it!